Heat Kernel estimates for some elliptic operators with unbounded diffusion coefficients

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June 2012, 32(6): 2285-2299. doi: 10.3934/dcds.2012.32.2285

Heat Kernel estimates for some elliptic operators with unbounded diffusion coefficients

Giorgio Metafune ^1, and Chiara Spina ^1,

1.	Dipartimento di Matematica “Ennio De Giorgi”, Università del Salento, C.P.193, 73100, Lecce, Italy, Italy

Received January 2011 Revised July 2011 Published February 2012

Abstract
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We prove heat kernel bounds for the operator $(1+|x|^\alpha)\Delta$ in $\mathbb{R}^N$, through Nash inequalities and weighted Hardy inequalities.

Keywords: Elliptic operators, Nash inequality, kernel estimates..

Mathematics Subject Classification: 47D07, 35B50, 35J25, 35J7.

Citation: Giorgio Metafune, Chiara Spina. Heat Kernel estimates for some elliptic operators with unbounded diffusion coefficients. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2285-2299. doi: 10.3934/dcds.2012.32.2285

References:

[1]	A. Bazan and W. Neves, The Caffarelli-Kohn-Niremberg's inequality for arbitrary norms, preprint.
[2]	A. Bazan and W. Neves, The Hardy and Caffarelli-Kohn-Niremberg inequalities revised, preprint, arXiv:1007.2005v1.
[3]	D. Bakry, F. Bolley, I. Gentil and P. Maheux, Weighted Nash inequalities, preprint, arXiv:1004.3456.
[4]	E. B. Davies, "One-Parameter Semigroups," London Mathematical Society Monographs, 15, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980.
[5]	E. B. Davies, "Heat Kernels and Spectral Theory," Cambridge Tracts in Mathematics, 92, Cambridge University Press, Cambridge, 1989.
[6]	K.-J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolutions Equations," Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000.
[7]	S. Fornaro and L. Lorenzi, Generation results for elliptic operators with unbounded diffusion coefficients in $L^p$- and $C_b$-spaces, Discrete and Continuous Dynamical Systems, 18 (2007), 747-772.
[8]	G. Metafune, E. M. Ouhabaz and D. Pallara, Long time behavior of heat kernels of operators with unbounded drift terms, J. Math. Anal. Appl., 377 (2011), 170-179. doi: 10.1016/j.jmaa.2010.10.023.
[9]	G. Metafune and C. Spina, Elliptic operators with unbounded diffusion coefficients in $L^p$ spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci., to appear.
[10]	G. Metafune and C. Spina, Kernel estimates for a class of Schrödinger semigroups, Journal of Evolution Equations, 7 (2007), 719-742. doi: 10.1007/s00028-007-0338-3.
[11]	G. Metafune, D. Pallara and M. Wacker, Feller semigroups on $\mathbb{R}^{N}$, Semigroup Forum, 65 (2002), 159-205. doi: 10.1007/s002330010129.
[12]	B. Muckenhoupt, Hardy's inequalities with weights, Studia Math., 44 (1972), 31-38.
[13]	B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc., 192 (1974), 261-274. doi: 10.1090/S0002-9947-1974-0340523-6.
[14]	E. M. Ouhabaz, "Analysis of Heat Equations on Domains," London Mathematical Society Monographs Series, 31, Princeton University Press, Princeton, NJ, 2005.
[15]	F.-Y. Wang, Functional inequalities and spectrum estimates: The infinite measure case, J. Funct. Anal., 194 (2002), 288-310. doi: 10.1006/jfan.2002.3968.

show all references

References:

[1]	A. Bazan and W. Neves, The Caffarelli-Kohn-Niremberg's inequality for arbitrary norms, preprint.
[2]	A. Bazan and W. Neves, The Hardy and Caffarelli-Kohn-Niremberg inequalities revised, preprint, arXiv:1007.2005v1.
[3]	D. Bakry, F. Bolley, I. Gentil and P. Maheux, Weighted Nash inequalities, preprint, arXiv:1004.3456.
[4]	E. B. Davies, "One-Parameter Semigroups," London Mathematical Society Monographs, 15, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980.
[5]	E. B. Davies, "Heat Kernels and Spectral Theory," Cambridge Tracts in Mathematics, 92, Cambridge University Press, Cambridge, 1989.
[6]	K.-J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolutions Equations," Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000.
[7]	S. Fornaro and L. Lorenzi, Generation results for elliptic operators with unbounded diffusion coefficients in $L^p$- and $C_b$-spaces, Discrete and Continuous Dynamical Systems, 18 (2007), 747-772.
[8]	G. Metafune, E. M. Ouhabaz and D. Pallara, Long time behavior of heat kernels of operators with unbounded drift terms, J. Math. Anal. Appl., 377 (2011), 170-179. doi: 10.1016/j.jmaa.2010.10.023.
[9]	G. Metafune and C. Spina, Elliptic operators with unbounded diffusion coefficients in $L^p$ spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci., to appear.
[10]	G. Metafune and C. Spina, Kernel estimates for a class of Schrödinger semigroups, Journal of Evolution Equations, 7 (2007), 719-742. doi: 10.1007/s00028-007-0338-3.
[11]	G. Metafune, D. Pallara and M. Wacker, Feller semigroups on $\mathbb{R}^{N}$, Semigroup Forum, 65 (2002), 159-205. doi: 10.1007/s002330010129.
[12]	B. Muckenhoupt, Hardy's inequalities with weights, Studia Math., 44 (1972), 31-38.
[13]	B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc., 192 (1974), 261-274. doi: 10.1090/S0002-9947-1974-0340523-6.
[14]	E. M. Ouhabaz, "Analysis of Heat Equations on Domains," London Mathematical Society Monographs Series, 31, Princeton University Press, Princeton, NJ, 2005.
[15]	F.-Y. Wang, Functional inequalities and spectrum estimates: The infinite measure case, J. Funct. Anal., 194 (2002), 288-310. doi: 10.1006/jfan.2002.3968.

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